The Drazin inverse of the sum of two matrices and its applications. (English) Zbl 1204.15017
The Drazin inverse of a square matrix \(A\), denoted by \(A^D\) is the unique matrix \(X\) that satisfies \(A^{k+1}X = A^k\), \(XAX = X\), \(AX=XA\), where \(k\) is the index of \(A\).
M. P. Drazin [Am. Math. Mon. 65, 506–514 (1958; Zbl 0083.02901)] proved that \((P+Q)^D = P^D + Q^D\), whenever \(PQ=QP=0\). Several generalizations of this result have been provided over the years. The authors present two other extensions, one of these under the condition \(PQP=PQ^2=0\). The expressions of the Drazin inverse are too complicated to be included here. Using these results, the authors also derive results for the Drazin inverses of certain block matrices.
M. P. Drazin [Am. Math. Mon. 65, 506–514 (1958; Zbl 0083.02901)] proved that \((P+Q)^D = P^D + Q^D\), whenever \(PQ=QP=0\). Several generalizations of this result have been provided over the years. The authors present two other extensions, one of these under the condition \(PQP=PQ^2=0\). The expressions of the Drazin inverse are too complicated to be included here. Using these results, the authors also derive results for the Drazin inverses of certain block matrices.
Reviewer: K. C. Sivakumar (Chennai)
MSC:
| 15A09 | Theory of matrix inversion and generalized inverses |
Citations:
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